Extremal phenomena in certain classes of totally bounded groups

<br>For various pairs <P,Q> of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly finer pseudocompact topological group topology and a proper dense pseudocompact subgroup: (1) G is O-dimensional and Abelian; (2) $G = H^α$ with α > ω, |H| > 1; (3) G is a dense subgroup of $T^{(ω⁺)}$.<br>Thwarting our attempts to improve (1), (2) and (3) are examples, for every α > ω, of pseudocompact groups G₀ and G₁ of weight α such that (a) there are surjective φ ∈ Hom(G₀,K) with K compact, φ continuous and open, and a dense, pseudocompact subgroup H of K such that $φ^{-1}(H)$ is not pseudocompact; and (b) G₁ admits no homomorphism onto any non-trivial product.